91影视

We can make three types of elementary matrices.

Row switching,$R_i鈫�R_j$

Notice that there is exactly one $2脳2$elementary matrix of type 1, that is,

$E_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

The matrix of T is of the form $\left[\begin{array}{cc}a& b\\ b& -a\end{array}\right]$, where $a^2+b^2=1$.Conversely, any matrix of this form represents a reflection of a line.

$E_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

As we know that represents the reflection of the line spanned by

$\left[\begin{array}{c}1\\ 1\end{array}\right]$

Row multiplication$R_i鈫�k{R}_{i}k鈮�0$

Since there are only two rows in $2脳2$matrices and so there are exactly two $2脳2$elementary matrices of type 2, that is,

$E_2=\left[\begin{array}{cc}k& 0\\ 0& 1\end{array}\right]$and$E_3=\left[\begin{array}{cc}1& 0\\ 0& k\end{array}\right]$

Clearly

$E_2=\left[\begin{array}{cc}k& 0\\ 0& 1\end{array}\right]$

represents a 鈥渟caling in $e_1$direction鈥� (leaving the $e_2$component unchanged) and

$E_3=\left[\begin{array}{cc}1& 0\\ 0& k\end{array}\right]$

represents a 鈥渟caling in $e_2$direction鈥� (leaving the $e_1$component unchanged)

Row addition,$R_i鈫�k{R}_j,k鈮�0$

Since there are only two rows in $2脳2$matrix and so there are exactly two $2脳2$elementary matrix of type 3, that are,

localid="1659345115910" $E_3=\left[\begin{array}{cc}1& 0\\ 0& k\end{array}\right]$and

The matrix of a horizontal shear is of the form $\left[\begin{array}{cc}1& k\\ 0& 1\end{array}\right]$, and the matrix of a vertical shear is of the form $\left[\begin{array}{cc}1& 0\\ k& 1\end{array}\right]$, where k is an arbitrary constant.

Therefore, $E_4=\left[\begin{array}{cc}1& k\\ 0& 1\end{array}\right]$represents horizontal shear and $E_5=\left[\begin{array}{cc}1& 0\\ k& 1\end{array}\right]$represents vertical shear.

There are only five $2脳2$elementary matrices.